No, you can't patent "drug that alleviates headaches", you can only patent a specific formulation, and a competitor is still free to create any other formulation that achieves the same result.
It's been attempted. They invent and patent X as a "method to do Y", and then try to sue others who do Y via some means other than X.
My point was that the original description of patents was not precise enough and implied that you could patent Y rather than X.
Still, patent claims are never as broad as "drug to alleviate headaches" or anything similar.
Edit: That is to say, I think you agree in principle, and you're finding points of disagreement in the wording that were not intended.
"mincing words" is a good description of law in general. That's no criticism at all.
This kind of super-broad patent enforcement happened with 3D printing, and is still happening with e-ink technology. In both cases, the patents didn't have centuries of prior art covering what would otherwise have been entire industries, and could protect their monopolies for 20 years.